3.22.97 \(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^2} \, dx\)

Optimal. Leaf size=151 \[ -\frac {(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}-\frac {1}{192} (65-1194 x) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{512} (3865-8082 x) \sqrt {3 x^2+5 x+2}+\frac {41053 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1024 \sqrt {3}}-\frac {1325}{128} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \]

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Rubi [A]  time = 0.10, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {812, 814, 843, 621, 206, 724} \begin {gather*} -\frac {(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}-\frac {1}{192} (65-1194 x) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{512} (3865-8082 x) \sqrt {3 x^2+5 x+2}+\frac {41053 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1024 \sqrt {3}}-\frac {1325}{128} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^2,x]

[Out]

-((3865 - 8082*x)*Sqrt[2 + 5*x + 3*x^2])/512 - ((65 - 1194*x)*(2 + 5*x + 3*x^2)^(3/2))/192 - ((34 + x)*(2 + 5*
x + 3*x^2)^(5/2))/(10*(3 + 2*x)) + (41053*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(1024*Sqrt[3])
- (1325*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/128

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 812

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + b*x + c*x^2)^p)/(e^2*(m + 1)*(m
+ 2*p + 2)), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
+ 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx &=-\frac {(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}-\frac {1}{8} \int \frac {(-332-398 x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx\\ &=-\frac {1}{192} (65-1194 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac {1}{768} \int \frac {(40938+48492 x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx\\ &=-\frac {1}{512} (3865-8082 x) \sqrt {2+5 x+3 x^2}-\frac {1}{192} (65-1194 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}-\frac {\int \frac {-2525724-2955816 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{36864}\\ &=-\frac {1}{512} (3865-8082 x) \sqrt {2+5 x+3 x^2}-\frac {1}{192} (65-1194 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac {41053 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{1024}-\frac {6625}{128} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {1}{512} (3865-8082 x) \sqrt {2+5 x+3 x^2}-\frac {1}{192} (65-1194 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac {41053}{512} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )+\frac {6625}{64} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {1}{512} (3865-8082 x) \sqrt {2+5 x+3 x^2}-\frac {1}{192} (65-1194 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac {41053 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{1024 \sqrt {3}}-\frac {1325}{128} \sqrt {5} \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 120, normalized size = 0.79 \begin {gather*} \frac {159000 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )+205265 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-\frac {2 \sqrt {3 x^2+5 x+2} \left (6912 x^5-28512 x^4-80064 x^3-118996 x^2+40412 x+293973\right )}{2 x+3}}{15360} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^2,x]

[Out]

((-2*Sqrt[2 + 5*x + 3*x^2]*(293973 + 40412*x - 118996*x^2 - 80064*x^3 - 28512*x^4 + 6912*x^5))/(3 + 2*x) + 159
000*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] + 205265*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6
+ 15*x + 9*x^2])])/15360

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IntegrateAlgebraic [A]  time = 0.85, size = 121, normalized size = 0.80 \begin {gather*} \frac {41053 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{512 \sqrt {3}}-\frac {1325}{64} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )+\frac {\sqrt {3 x^2+5 x+2} \left (-6912 x^5+28512 x^4+80064 x^3+118996 x^2-40412 x-293973\right )}{7680 (2 x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^2,x]

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(-293973 - 40412*x + 118996*x^2 + 80064*x^3 + 28512*x^4 - 6912*x^5))/(7680*(3 + 2*x)) +
(41053*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[3]*(1 + x))])/(512*Sqrt[3]) - (1325*Sqrt[5]*ArcTanh[Sqrt[2 + 5*x +
3*x^2]/(Sqrt[5]*(1 + x))])/64

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fricas [A]  time = 0.43, size = 149, normalized size = 0.99 \begin {gather*} \frac {205265 \, \sqrt {3} {\left (2 \, x + 3\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 159000 \, \sqrt {5} {\left (2 \, x + 3\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 4 \, {\left (6912 \, x^{5} - 28512 \, x^{4} - 80064 \, x^{3} - 118996 \, x^{2} + 40412 \, x + 293973\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{30720 \, {\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^2,x, algorithm="fricas")

[Out]

1/30720*(205265*sqrt(3)*(2*x + 3)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 15900
0*sqrt(5)*(2*x + 3)*log(-(4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) - 124*x^2 - 212*x - 89)/(4*x^2 + 12*x + 9)
) - 4*(6912*x^5 - 28512*x^4 - 80064*x^3 - 118996*x^2 + 40412*x + 293973)*sqrt(3*x^2 + 5*x + 2))/(2*x + 3)

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giac [B]  time = 1.17, size = 671, normalized size = 4.44 \begin {gather*} -\frac {41053}{3072} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}{{\left | 2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {1325}{128} \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {325}{128} \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {1304805 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{9} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 2064120 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{8} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 4382950 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{7} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 10490640 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{6} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 19083456 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 33372000 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{4} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 42760170 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 60102000 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{2} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 21448395 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 36498600 \, \sqrt {5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{7680 \, {\left ({\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{2} - 3\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^2,x, algorithm="giac")

[Out]

-41053/3072*sqrt(3)*log(abs(-2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + 2*sqrt(5)/(2*x + 3))/abs(2
*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + 2*sqrt(5)/(2*x + 3)))*sgn(1/(2*x + 3)) + 1325/128*sqrt(5
)*log(abs(sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3)) - 4))*sgn(1/(2*x + 3)) - 325/12
8*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3)*sgn(1/(2*x + 3)) + 1/7680*(1304805*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2
+ 3) + sqrt(5)/(2*x + 3))^9*sgn(1/(2*x + 3)) - 2064120*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt
(5)/(2*x + 3))^8*sgn(1/(2*x + 3)) - 4382950*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^7*sgn
(1/(2*x + 3)) + 10490640*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^6*sgn(1/(2*x + 3
)) + 19083456*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^5*sgn(1/(2*x + 3)) - 33372000*sqrt(
5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^4*sgn(1/(2*x + 3)) - 42760170*(sqrt(-8/(2*x +
3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^3*sgn(1/(2*x + 3)) + 60102000*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x
+ 3)^2 + 3) + sqrt(5)/(2*x + 3))^2*sgn(1/(2*x + 3)) + 21448395*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt
(5)/(2*x + 3))*sgn(1/(2*x + 3)) - 36498600*sqrt(5)*sgn(1/(2*x + 3)))/((sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3)
+ sqrt(5)/(2*x + 3))^2 - 3)^5

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maple [A]  time = 0.06, size = 195, normalized size = 1.29 \begin {gather*} \frac {1325 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{128}+\frac {41053 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\right )}{3072}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{10 \left (x +\frac {3}{2}\right )}-\frac {53 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{20}+\frac {199 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{192}+\frac {1347 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{512}-\frac {265 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{48}-\frac {1325 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{128}+\frac {13 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(5/2)/(2*x+3)^2,x)

[Out]

-13/10/(x+3/2)*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-53/20*(-4*x+3*(x+3/2)^2-19/4)^(5/2)+199/192*(6*x+5)*(-4*x+3*(x+3/
2)^2-19/4)^(3/2)+1347/512*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)+41053/3072*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(-
4*x+3*(x+3/2)^2-19/4)^(1/2))-265/48*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-1325/128*(-16*x+12*(x+3/2)^2-19)^(1/2)+1325/
128*5^(1/2)*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))+13/20*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4
)^(5/2)

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maxima [A]  time = 1.46, size = 163, normalized size = 1.08 \begin {gather*} -\frac {1}{20} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {199}{32} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {65}{192} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{4 \, {\left (2 \, x + 3\right )}} + \frac {4041}{256} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {41053}{3072} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {1325}{128} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {3865}{512} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^2,x, algorithm="maxima")

[Out]

-1/20*(3*x^2 + 5*x + 2)^(5/2) + 199/32*(3*x^2 + 5*x + 2)^(3/2)*x - 65/192*(3*x^2 + 5*x + 2)^(3/2) - 13/4*(3*x^
2 + 5*x + 2)^(5/2)/(2*x + 3) + 4041/256*sqrt(3*x^2 + 5*x + 2)*x + 41053/3072*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 +
5*x + 2) + 3*x + 5/2) + 1325/128*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2
) - 3865/512*sqrt(3*x^2 + 5*x + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^2,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+5*x+2)**(5/2)/(3+2*x)**2,x)

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/(4*x**2 +
12*x + 9), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-113*x**3*sqrt(3
*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) -
Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x)

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